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Abstract

Background

Maternally transmitted symbionts have evolved a variety of ways to promote their spread
through host populations. One strategy is to hamper the reproduction of uninfected
females by a mechanism called cytoplasmic incompatibility (CI). CI occurs in crosses
between infected males and uninfected females and leads to partial to near-complete
infertility. CI-infections are under positive frequency-dependent selection and require
genetic drift to overcome the range of low frequencies where they are counter-selected.
Given the importance of drift, population sub-division would be expected to facilitate
the spread of CI. Nevertheless, a previous model concluded that variance in infection
between competing groups of breeding individuals impedes the spread of CI.

Results

In this paper we derive a model on the spread of CI-infections in populations composed
of demes linked by restricted migration. Our model shows that population sub-division
facilitates the invasion of CI. While host philopatry (low migration) favours the
spread of infection, deme size has a non-monotonous effect, with CI-invasion being
most likely at intermediate deme size. Individual-based simulations confirm these
predictions and show that high levels of local drift speed up invasion but prevent
high levels of prevalence across the entire population. Additional simulations with
sex-specific migration rates further show that low migration rates of both sexes are
required to facilitate the spread of CI.

Conclusion

Our analyses show that population structure facilitates the invasion of CI-infections.
Since some level of sub-division is likely to occur in most natural populations, our
results help to explain the high incidence of CI-infections across species of arthropods.
Furthermore, our work has important implications for the use of CI-systems in order
to genetically modify natural populations of disease vectors.

Background

Vertically transmitted symbionts are common in nature. They are of diverse phylogenetic
origin, comprising viruses, bacteria, protozoa, and fungi, and reside in an equally
diverse range of hosts, including both animals and plants[1-5]. Many vertically transmitted symbionts live inside the host cytoplasm and accordingly
their transmission is strictly maternal. For these symbionts to spread through a host
population, the number of infected daughters produced by an infected female must exceed
the number of daughters produced on average in the population. Symbionts have evolved
a number of strategies to achieve this goal. Some, such as secondary symbionts of
aphids, are beneficial and increase the host's overall fitness through a positive
effect on fecundity or survival [6,7]. Others spread by biasing the sex ratio of their host towards females. This strategy
is common in symbionts of arthropods, which achieve sex ratio distortion by inducing
parthenogenesis, feminizing genetic males, or selectively eliminating male hosts [2,8]. A third strategy differs fundamentally from the previous two in that it promotes
infection indirectly by impeding the reproduction of uninfected female hosts rather
than affecting that of infected ones [8-10]. This is achieved through a mechanism called cytoplasmic incompatibility (CI) which
appears to have evolved at least twice independently in two bacterial symbionts of
arthropods, Wolbachia [11] and Cardinium [12]. Incompatibility is presumably caused by a modification in the sperm of infected
males that causes increased zygote mortality unless rescued by a symbiont present
in the egg [2]. CI therefore specifically targets the fecundity of uninfected hosts.

Cytoplasmic incompatibility benefits infected individuals by reducing the fecundity
of uninfected females in a targeted way. This reduction increases the relative fitness
of infected females, i.e., their fecundity measured relative to the average fecundity
in the population. The degree to which CI increases the relative fitness of infected
depends on the frequency of incompatible matings, making selection on CI frequency-dependent
[9,13-16]. Whether CI is selectively favored depends on the balance between the increase in
relative fitness of infecteds due to CI and forces that act against its spread, namely
a fecundity cost of bearing an infection and imperfect transmission of the symbiont.
In the presence of a cost of infection and/or imperfect transmission, an unstable
infection equilibrium exists (often referred to as the 'invasion threshold'), separating
the lower frequency range in which infection is counter-selected and the higher frequency
range where it is selected for.

For symbionts to spread in a host population, infection frequency has to exceed the
unstable equilibrium. Many authors have therefore made the verbal argument that random
genetic drift would be required for the successful spread of CI-symbionts [9,17]. Egas et al. [17] tested this conjecture in computer simulations that determined the probability of
CI spread in random mating populations of different sizes (and hence subject to varying
levels of genetic drift). The authors concluded that random drift was unlikely to
drive infection frequency beyond the invasion threshold and allow it to go to fixation.
This result is at odds with the high number of species that have been found to be
infected with CI-inducing parasites [see [18] for a review]. Egas et al. therefore conclude that in addition to CI, infection spread
is driven by factors such as other reproductive manipulations, effects of infection
on the population sex ratio or other fitness-compensating effects [17].

The conclusion of Egas et al.'s seems pessimistic for the spread of CI infection in
panmictic populations of large size. However, their data also show that CI can spread
with considerable probabilities when populations are small. This raises the possibility
that the spread of CI might be facilitated if a large population is subdivided into
a number of discrete demes that are linked by migration. Population sub-division is
known to induce increased levels of local genetic drift, which could drive infection
frequency beyond the invasion threshold within demes. Despite its clear relevance,
the effect of genetic drift on CI spread in sub-divided population has found little
treatment in the theoretical literature. Some studies have analyzed CI-dynamics either
in continuous [19] or discrete space [20], but have done so using deterministic models which ignore the stochastic effect due
to drift. The only model including drift (and only implicitly) has come to the surprising
prediction that population structure would impede the spread of infection. In this
model, Wade and Stevens [21] assume that hosts aggregate in groups to mate and reproduce before being mixed and
compete at the scale of the whole metapopulation. Frequency change of CI was then
calculated as a function of the variance in infection frequency between breeding groups.
Wade and Stevens showed that increasing this variance, as would happen because of
drift or if infection appeared in a subset of the groups, raised the invasion threshold
and reduced the per-generation change in average infection frequency across groups.

Motivated by the discrepancy between biological intuition and theoretical prediction,
we present here a new analysis of CI dynamics in sub-divided host populations. We
first derive an analytical model which illustrates the different components of the
selection pressure acting on a CI phenotype and how they are affected by population
sub-division. This model allows us to derive the conditions under which a rare infection
can spread in a population and to determine the expected equilibrium infection frequency.
We complement the analytical model with individual-based simulations. These relax
some of the simplifying assumptions made in the model and generate predictions for
more complex population structures.

Results

Analytical model

Similar to previous models treating the panmictic case we will ignore the dynamics
of symbiont populations within individual hosts and concentrate on the presence or
absence of parasite infection. This allows us to conveniently model the spread of
infection as a two state model in which individuals are in one of the two states 'infected'
and 'uninfected'. Transmission of infection state is purely maternal, resembling symbiont
transmission in natural host populations. Vertical transmission of infection is assumed
to be perfect and all offspring of infected mothers are infected. This assumption
is justified because natural parasites causing CI usually show high rates of vertical
transmission [9], but is relaxed in simulations presented below.

We assume that hosts live in a population composed of an infinite number of demes,
each containing a finite number N of breeding males and N females (i.e., infinite island model assumptions). Hosts are diploid and semelparous
(no overlapping generations) and undergo the following lifecycle: (1) Adult hosts
mate randomly within demes. (2) Adult hosts reproduce. Each female produces a large
number of offspring that is a function of the infection status of her and her mating
partner. The fecundity of an uninfected female mated to an infected male (incompatible
mating), relative to that of uninfected female mated to an uninfected male, is (1
- B) and the relative fecundity of an infected female is (1 - C), independently of the genotype of her mate. In these fecundities, C is the fecundity cost of infection borne by infected females and B is the fecundity cost of cytoplasmic incompatibility borne by uninfected females mated
to infected males (note that for clarity we have chosen to deviate from the usual
notation in which C = sf and B = sh). All adults die after reproduction. (3) Juvenile hosts disperse. Male and female
juveniles disperse with equal probabilities and randomly, meaning that any non-natal
deme is reached with same probability. The probability that a juvenile is sampled
in its natal deme after dispersal is given by (1 - m) where m is the migration rate. (4) Juveniles compete for access to reproduction. Competition
is sex-specific and occurs within demes. In each deme, N juveniles of each sex reach adulthood, the remainder die. This life cycle drastically
differs from that of Wade & Stevens [19], where mixing of individuals is complete after breeding and occurs at the scale of
the whole population. However, our model converges towards that of Wade & Stevens
at the higher migration limit m → 1.

Assuming this life-cycle and weak selection (small C and B), we show in the Appendix that the change in frequency (Δp) of infection in the population can be written as

(1)

where Fst, , and g(N, m) are terms describing the effect of population sub-division on the probability that
individuals in a deme share their infection status. More precisely, Fst is Wright's measure of population structure describing the probability that two randomly
sampled individuals within a deme share their infection status by descent, is the probability that one male and two females sampled with replacement from the
same deme share their infection status by descent, and g(N, m) is a function depending on deme size and migration rate (see eq. 21 in the Appendix).

The selective pressure on infection (the term in square brackets) comprises three
components. The first, -C(1-Fst), describes the effect of directional selection against infection due to the fecundity
cost C borne by infected females. The fecundity cost is weighted by the probability (1-Fst). This weighting expresses the fact that population sub-division reduces the strength
of this component of selection. Specifically, genetic drift leads to increased coalescence
within demes, thereby homogenising local demes with respect to infection status and
levelling differences in fecundity between competitors. The second and third components
of selection capture the change in infection frequency due to the benefit of CI. These
arise whenever an infected female enjoys increased relative fitness due to the reduced
productivity of an uninfected females having an incompatible mating with an infected
male. Accordingly, the benefit of CI is proportional not only to the cost of incompatibility,
B, but also to the frequency with which one infected female, one infected male and
one uninfected female occur in a same deme (the first being the beneficiary, the latter
two the incompatible mates). Equation 1 distinguishes two cases that can lead to this
constellation, based on the genetic relationship between the infected female and male.
The first of the benefit terms, B(1-m)2(Fst - ), is frequency-independent (does not contain p) and measures the beneficial effect arising from events where the infected male and
female are related, i.e. are both infected due to recent coalescence. Accordingly,
the term weights the cost of incompatibility, B, by the term (FST - F3R) which quantifies the joint probability that within a deme, a female shares her infection
status by recent descent with an infected male mated to an uninfected female (who
will suffer from incompatibility). Local genetic drift affects the term (FST - F3R) in a non-trivial way, because on one side it increases the probability that a female
shares the same infection status as a male, but on the other side it decreases the
probability that this male will have mated with a female bearing the alternative status.
As a consequence of these opposing effects, (FST - F3R) is not necessarily a monotonic function of N and m. The weighting by (1 - m)2 expresses the fact that the offspring of an infected female have to remain in their
natal patch and compete against other sedentary offspring in order for an increase
in the relative fitness of infected females to arise. The second benefit term, pB(1-g(N, m)), measures the benefit of CI that is independent of recent coalescence. Here, the
infected male and female are unrelated in the sense there is no recent coalescence
in their infection status. Accordingly, the probability of such an event is frequency-dependent
and the term includes the average frequency p of infection in the population. The term is further offset by a quantity g(N, m) which measures the effect of population sub-division on the frequency with which
such a benefit occurs. The function g(N, m) decreases monotonically in N and m (see Appendix), reflecting the fact that with reduced population sub-division, the
benefit of CI relies more and more on interactions between unrelated individuals.

Equation 1 also allows us to recover the dynamics of CI infections considered by previous
treatments. When deme size becomes very large (N → ∞) or when migration is complete (m → 1), equation 1 reduces to

Δp = p(1 - p) [-C + pB],(2)

which is the weak-selection equivalent of the frequency change calculated by [14].

We will now consider the situation when infection is rare (p → 0). This situation is highly relevant, because it is the natural condition faced
by all invading CI-infections and the dynamics of infection at this point determine
the probability of invasion (or, in mathematical terms, the stability of the equilibrium
p = 0). Taking the lower frequency limit of equation 1, we can derive the condition
for the invasion of a rare CI-infection in terms of the cost-to-benefit ratio C/B,

(3)

The expression on the right-hand size of this inequality is a function of deme size
(N) and migration rate (m). Equating both sides of inequality 3, we can define a threshold cost-to-benefit
ratio above which infections will not be able to invade the population. Figure 1 shows how this threshold ratio varies with both host migration rate and deme size.
For population with very large deme sizes and very high migration rates (i.e., quasi-panmixia)
the threshold approaches zero and infections cannot invade unless they are virtually
cost-free relative to the intensity of incompatibility. With decreasing migration
rate, however, the threshold ratio increases monotonically, indicating that infections
can spread even if associated with a fecundity cost (C>0) and/or incomplete incompatibility
(B<1). The change of the threshold ratio with deme size is non-monotonous, the threshold
ratio being highest at intermediate deme sizes. This effect can be interpreted in
terms of the impact of local drift on the frequency of incompatible matings. In host
populations with intermediate deme size, drift maintains the local variance in infection
at a level that maximizes the benefit of incompatibility, making such population most
prone to the invasion of CI-infections.

Simulations

To complement the analytical model, we ran individual-based computer simulations of
CI spread, based on an extended version of the population genetics software Nemo [22]. The simulations allowed us to relax some of the more idealistic assumptions of the
model and measure additional parameters of infection dynamics, such as the speed of
spread and the equilibrium frequency attained.

The simulations considered a finite population of 1800 individuals (half males, half
females) subdivided into of nd demes of N individuals each. The demes are linked by migration at rate m. We implemented two migration models, an island model (all demes equidistant) and
a stepping stone model (populations arranged on a ring). Furthermore, in one set of
simulations we explored the effect of differential male and female migration in an
island model.

The life cycle is identical to that assumed in the analytical model. Male and female
fecundity was assumed Poisson distributed and each individual was assigned a number
of offspring drawn form a distribution with mean and variance f = 10. The cost of infection was implemented by reducing the fecundity of infected
females by a factor (1 - C). Cytoplasmic incompatibility acted upon fertilization. An uninfected egg fertilized
by the sperm of an infected father died with probability B. In all simulations, we relaxed the assumption of perfect vertical transmission made
in the analytical model. Thus, we assumed that infection was lost with a probability
μ from mother to offspring.

Throughout this study we used a standard set of infection parameters with an incompatibility
cost of B = 0.5, a fecundity cost of C = 0.01, and a rate of parasite loss during vertical transmission of μ = 0.05. Based
on the model for an infinite panmictic population [14], an infection with these parameters is selectively favoured when its frequency exceeds
the invasion threshold of pit = 0.13 and will attain an equilibrium frequency of pe = 0.94. The parameter values used are inspired by empirical data obtained in populations
of Drosophila simulans harbouring a CI-Wolbachia [23]. Additional simulations (data not shown) showed that varying the values has the effect
expected under the deterministic model of infection dynamics with panmixia.

At the start of each simulation, one deme was inoculated with one infected female
and one infected male. This starting frequency is well below the invasion threshold
of pit = 0.13 (and hence counter-selected) when considering the whole population of 900 males
and 900 females, as well as when considering deme sizes that exceed 7 males and 7
females. Simulations were run for 2000 generations and infection frequency was recorded
for each deme at intervals of 10 generations. We considered an infection to have invaded
if it was still present at the end of the simulation run (generation 2000). Simulations
were replicated 1000 times for any combination of parameters.

Effects of dispersal rate and deme size in an island model

In a first set of simulations, we analyzed the dynamics of infection as a function
of host migration rate and deme size in an island model of population structure. In
these simulations, the population was sub-divided into demes of sizes N = {4, 6, 12, 20, 60, 90, 180}. Since the sex ratio was assumed to be even in our simulations,
these deme sizes are equivalent to 2, 3, 6, 10, 30, 45, and 90 reproductive females.
With a total population size of 1800, the array of deme sizes imply deme numbers of
nd = {450, 300, 150, 90, 30, 20, 10}. Demes were linked by male and female migration
at equal rates. The range of migration rates used was m = {0.01, 0.05, 0.1, 0.2}. We ran simulations for all combinations of migration rate
m and deme size N.

Frequencies of invasion (i.e., the proportions of replicate simulation runs in which
the infection was present after 2000 generations) varied with both deme size and migration
in a way that fit the predictions of our analytical model (Fig. 2). While increasing host migration rate lead to a decline in invasion frequency (Fig.
2A), infections were more likely to persist when deme size was intermediate (Fig. 2B). Figure 2 shows that population sub-division (restricted migration) allows CI infection to
invade populations it would not have invaded under panmixia. For example, with deme
sizes of N = 180, infection can sometimes invade with migration rates of m = 0.01 and m = 0.05, but not with m = 0.1 and above, which for ten demes corresponds to panmixia.

In addition to its effect on the probability of spread, population sub-division also
influenced the equilibrium frequency they attained. Generally, infections attained
a higher equilibrium frequency (prevalence) under conditions that minimized the amount
of local genetic drift. Thus, prevalence in populations with strong drift (low migration
rate and/or small deme size) remained considerably lower than the value predicted
for panmixia (pe = 0.94) at which populations with weak drift (large demes and/or high rates of migration)
stabilized (Fig. 3). The low prevalence in high-drift populations is not due to a failure to attain
equilibrium. Rather, low prevalence is the result of two mechanisms. First, drift
causes prevalence to fluctuate around the equilibrium of pe = 0.94. Because infection frequency cannot exceed 100%, large stochastic fluctuations
in populations with strong drift will tend to decrease mean prevalence. Second, under
conditions of strong local drift, population-wide mean prevalence is lowered by the
total loss of infection in some demes (Fig. 3). For instance, in the two cases showing the lowest mean prevalence (N = 12 with m = 0.01, and N = 6 with m = 0.05), around a quarter of demes had lost infection while the prevalence in the
remaining infected demes was 86–88% (compared to a population-wide prevalence of about
64%). Additional simulations have shown that such polymorphism is stable over large
time-spans (more than 5,000 generations).

Figure 3.Simulation results on the mean population prevalence and between-deme polymorphism
in infection. Mean population prevalence (± SE) and the proportion of demes infected as a function
of deme size in simulations with equal migration rate in males and females. Results
are shown for host migration rates of m = 0.01 (solid line), m = 0.05 (dashed line), m = 0.1 (dotted line), and m = 0.2 (dashed-dotted line). The grey line indicates the equilibrium frequency of pe = 0.94 predicted for a panmictic population of infinite size.

Effects of dispersal rate and deme size in a stepping stone model

The island model used in the analytical model and the above simulations is convenient
from a modelling point of view, but is representative of few, if any, natural situations.
Indeed, it assumes that all demes of a population are equidistant, while in natural
populations, demes tend to be arranged in a geographic pattern that causes some pairs
of demes to be further apart than others. Accordingly, some demes are more likely
to exchange migrants than others. In order to assess the influence of geography on
the spread of CI, we performed a set of simulations in a simple stepping stone setting,
in which the demes were arranged on a circle. Migration occurred between neighbouring
demes along the circle. Simulations were run for deme sizes of N = {4, 6, 12, 20,
60, 180} (deme numbers nd = {450, 300, 150, 90, 30, 10}) and migration rates of m = {0.01, 0.05, 0.1, 0.2, 0.4, 0.6}.

Figure 4 shows the frequency of invasion observed in these simulations as a function of migration
rate (4A) and deme size (4B). Overall, the results are qualitatively equivalent to
those obtained under the island model. Thus, the frequency of CI-invasion tends to
decrease with increasing host migration rate. Furthermore, invasion frequency is highest
for intermediate deme size. However, comparing Figures 2 and 4 shows that there are quantitative differences between the island model and the stepping
stone model. In particular, the explicit geographical arrangement of demes in the
stepping stone model appears to increase the probability of CI spread with high rates
of host migration. This difference could arise because the limiting migration to adjacent
demes in the stepping stone model reduces the degree to which infection is diluted
across the entire population, making it more likely that the infection frequency will
remain above the infection threshold in groups of neighbouring demes.

Infection dynamics as a function of male and female dispersal rates

In a third set of simulations, we addressed the relative importance of male and female
dispersal rates for the spread of CI infections. We ran simulations for all combinations
of male and females migration rates of m = {0.01, 0.05, 0.1, 0.2}. Population sub-division was fixed to nd = 30 and N = 60.

The parasites being maternally transmitted, female dispersal was a major determinant
of infection dynamics. As shown in Figure 5, the frequency of CI invasion decreased with female dispersal rate. However, male
migration rate also had an effect, in that female philopatry favoured the spread of
infection more when males dispersed at a low or moderate rate and less if male dispersed
at a high rate.

Discussion

In this paper we have investigated the effect of host population structure on the
spread of incompatibility-inducing symbionts. Our analyses demonstrate that population
structure can greatly facilitate the spread of CI. In our simulations, infections
spread from being present in a single individual of each sex (pf = pm = 1/900) to an equilibrium frequency of p ≈ 0.94 with probabilities as high as 15% (cf. Figs. 2 and 4), despite the relatively benign level of incompatibility used (B = 0.5). These results suggest that the invasion threshold predicted by panmictic models
does not represent a major barrier to invasion of CI into structured populations.
If regular opportunities exist for horizontal transmission of infections into the
populations, the spread of infection is only a matter of time. Population sub-division,
which can be assumed to affect most natural populations to some degree, could thus
provide the key to understanding the high incidence of CI-inducing symbiont infections
that has been observed in arthropods, and in particular insects [18].

Both the model and the simulations consistently showed that the rate of invasion of
CI-symbionts varies according to both parameters of host population structure considered
in our study, deme size and migration rate. The latter of these parameters affects
infection in a straightforward way; high levels of host migration will result in the
dilution of infection out of a deme into the population at large and therefore decrease
the probability of successful invasion. Furthermore, CI spread relies on limited dispersal
of both males and females, in order to assure that infected females enjoy an increased
relative fitness as a result of uninfected females suffering incompatible matings.
The other parameter, deme size, shows an interesting non-monotonous effect whereby
CI-infections invade easiest when deme size is intermediate. Under these conditions,
local genetic drift creates heterogeneity in infection among the local breeding population
that guarantees a high proportion of incompatible matings and hence a large selective
advantage for infection. Small deme size does not favour the spread of infection for
two reasons. First, strong drift results in the frequent elimination of infection
shortly after the initial inoculation of a population. Second, in the event that infection
is not lost but fixed in one deme, it is subject to negative between-deme selection
because the fecundity cost of infection reduces the productivity of infected demes.
Large deme sizes, in turn, are disadvantageous because local drift is not strong enough
to overcome the fitness disadvantage that CI-infections suffer at low frequencies
due to the cost of infection.

As outlined above, the non-monotonous dependence of invasion probability on deme size
results partly from selection against demes fixed for either the uninfected or the
infected state. Infected demes lose out in between-deme competition because their
productivity is reduced by the cost of infection. Accordingly, the low invasion success
of a CI strategy in populations with small demes is related to the fact that infection
reduces fecundity irrespectively of whether CI results in harm to uninfected individuals
or not. Invasion success in small demes is expected to be higher for agents (be it
genes or symbionts) in which the fecundity cost is conditional on the expression of
the harming act (such as CI). This is the situation usually considered in models of
the evolution of spiteful behaviour [24,25]. In these models, invasion of spiteful traits has been found to decrease monotonously
with deme size.

The simulation results presented here demonstrate not only that population sub-division
affects CI-dynamics, but also that this effect varies according to the population
model considered. Thus, host migration had a stronger negative effect on CI invasion
in the island model than in the stepping-stone model. This quantitative difference
can be explained by a lesser degree of dilution of infection in the geographically
explicit stepping stone model. Thus, local migration in this population setup resembles
diffusion and is expected to result in a spatial correlation of infection frequency,
with adjacent demes being more similar in infection than demes further apart. Accordingly,
migration out of infected demes will drive up infection frequency in adjacent demes
very effectively and is more likely to bring infection frequency in the range of positive
selection for infection. The flipside of localized migration is, however, that the
spread of infection across the population as a whole proceeds more slowly and more
generations are needed to reach the equilibrium prevalence in the population as a
whole (data not shown).

Our prediction that population sub-division can favour the spread of a CI infections
differs diametrically from that of Wade and Stevens [21]. These authors found that population structure impedes the spread of CI. A closer
look at the two models reveals that this discrepancy is rooted in the different ways
in which population structure is implemented in the life-cycle underlying the models.
Wade and Stevens' model assumes that reproduction takes place within groups. Offspring
produced in different matings groups then undergo complete dispersal and compete at
the level of the entire population [hard selection sensu [26]], before being partitioned in groups again for the next round of reproduction. Population
structure is implemented as variance in infection frequency between mating groups,
which is a fraction of the total variance in infection frequency p(1-p). Importantly, the portion of the total variance in infection that is attributed
to the between-deme component is arbitrarily determined by a parameter. Accordingly,
between- and within-deme variance are both assumed to vary completely independently
of the migration rate and are thus independent of the geographical structure of the
population. In this setting, increasing the variance in infection between mating groups
will necessarily be disadvantageous for the spread of infection, because it is associated
with a more homogeneous infection within demes and hence a reduced frequency of incompatible
matings. The spread of CI is further compromised by the assumption that competition
occurs at the level of the entire population. Global competition reduces the net advantage
of CI because fitness is measured by comparing the fecundity of infected females to
the population average rather than the deme average. As a consequence, a high frequency
of incompatible matings within some individual demes will only generate a small increase
in the relative fitness of infected females across the entire population. In our model,
population structure is not imposed as an external parameter but arises naturally
as a consequence of limited dispersal between permanent demes. Both reproduction and
competition take place within demes, but separated by an episode of dispersal. With
such a setting, intermediate between hard and soft selection, population structure
affects both the frequency of incompatible matings and the probability that infected
individuals will benefit from the increase in relative fitness brought about by CI.
Our model shows that, if implemented in this natural way, population sub-division
is conducive to the spread of CI-symbionts because the benefit of CI is likely to
go to infected individuals rather than being diluted in the population at large. Under
the assumption that competition occurs before dispersal (pure soft selection), we
expect that population sub-division would be even more in favor of the spread of CI
than under our life cycle because competition would occur only locally, thus increasing
the benefits of reducing the fecundity of neighbouring uninfected females. This suggests
that our results are qualitatively robust to changes of assumptions on the timing
of dispersal.

The analytical model presented here assumes that individuals perform a large number
of random matings within demes. Violations of this assumption due to systematic inbreeding
or inbreeding avoidance alter the dynamics of an incompatibility-inducing mutant.
Inbreeding reduces the expected number of incompatible matings and hence impedes the
spread of CI. Outbreeding, on the contrary, increases the number of incompatible matings
and promotes the spread of infection [27]. These effects of the mating system are expected to occur irrespective of population
structure and hence add to the effects of dispersal rate and deme size predicted here.
However, population structure is expected to have an influence on the quantitative
difference between the frequency change with outbreeding/inbreeding and that with
random mating. Both the advantage of outbreeding and the disadvantage of inbreeding
are subject to the presence of diversity in infection within the mating populations.
Under conditions of strong drift, the mating system can therefore be expected to have
a small effect on infection dynamics.

In addition to a simple mating system, random mating, our model and simulations assumed
non-overlapping generations. While this assumption might hold for some real-world
cases, the majority of species show some degree of overlap between generations. Population
age structure has been suggested to negatively affect the dynamics of CI and impeding
infection spread [28], although the assumptions of this model are unclear. We have not extended our model
in this direction but we would expect that including age structure would have a positive
effect on the probability of CI spread. With overlapping generations, only a fraction
of the breeding population is replaced at every generation. As a consequence, increasing
age structure will, for a given deme size, reduce the part of the breeding population
that has just reached adulthood and potentially migrated. Thus, age structure will
tend to reduce the effective migration rate and accentuate population sub-division,
thereby increasing the probability of CI spread (cf. Fig. 2A).

The predictions generated by our model and simulations have several implications for
empirical research on CI-inducing symbionts. First, they generate the testable prediction
that symbionts causing CI should be relatively more frequent in species or populations
that show a higher degree of sub-division. Furthermore, our results on overall prevalence
(Fig. 3) highlight potential problems for obtaining meaningful estimates of prevalence. Most
natural hosts populations will be sub-divided and potentially subject to strong genetic
drift. Accordingly, the effective population size was found to be around ten times
smaller than the numerical population size in a large survey of data on wild animal
populations [29]. Under conditions of local drift, measures of prevalence should be based on screens
for infection in several demes, because strong drift can generate local variation
in infection frequency. A wide sampling screen is the more important the lower the
migration rate of hosts and the smaller the size of local breeding populations.

Finally, our results are of fundamental importance with regards to the exploitation
of CI as a drive-systems for genetic manipulation of natural populations of arthropods.
Introducing transgenes into natural populations can be desirable for a number of reasons,
the most important of which is the aim of controlling the transmission of vector-borne
disease such as malaria. The realization that CI can bring about the rapid spread
of symbiont infection in large host populations has led to the proposition of using
the power of this system to drive transgenes through a host population [e.g., [30,31]]. However, previous analyses of the dynamics of such a system based on a panmictic
model suggested that large numbers of genetically modified organisms would be required
in order to overcome the invasion threshold to achieve successful spread of the construct
[31]. Our results suggest that this limitation applies to a much lesser degree to situations
in which vector populations are sub-divided. As shown in Figures 2 and 4, infections in highly subdivided populations spread with reasonable probability,
even if populations are inoculated with a single pair released in one deme. The probability
of successful invasion is expected to be much increased if the number of released
individuals is large relative to the size of local demes. By reversing their logic,
the predictions of our model further suggest that invasion success could be increased
by artificially limiting host migration. A combined strategy of local release of transgenic
hosts combined with, for instance, the pesticide treatment of adjacent demes would
help the establishment of infection in the focal deme, creating a base for subsequent
spread through the remainder of the population.

Methods

All methods are described in the main text. Additional information on the derivation
of the analytical model is provided in the Appendix (Additional file 1).

Additional file 1. The Appendix contains a detailed derivation of the analytical model.

Conclusion

Our work has shown that population sub-division greatly facilitates the spread of
CI-inducing symbionts and allows infections to invade large populations, even when
initially rare. Our study therefore helps to reconcile the high incidence of CI-infections
across arthropod species with the considerable barriers to invasion predicted by models
of CI-dynamics based on large panmictic populations. Furthermore, our predictions
validate the use of CI-systems for the genetic manipulation of natural populations.
Indeed, sub-divided populations of hosts could be successfully infected with relatively
small numbers of genetically engineered CI-agents, in particular if gene flow can
be artificially restricted during the early periods of invasion.

Authors' contributions

MR conceived the research, MR, LL, and FG performed the research. MR, LL, and FG wrote
the paper.

Acknowledgements

We would like to thank Sylvain Charlat, Olivier Duron, Jan Engelstädter, Greg Hurst,
Andrew Pomiankowski, Michael Whitlock and several anonymous referees (in particular
one with a sense for coalescence theory) for helpful comments on the manuscript of
this article. MR was supported by a Marie Curie Intra-European fellowship from the
European Commission, FG and LL by fellowships from the Swiss National Science Foundation.
FG also benefited from a NSERC (Canada) grant to M. Whitlock. Simulations were performed
on Vital-IT http://www.vital-it.ch and Westgrid http://www.westgrid.ca. The software and its source are available at http://nemo2.sourceforge.net.